Modelling Invasive Processes in

Biology

Philip K Maini

Wolfson Centre for Mathematical Biology

Mathematical Inst

Oxford

UK

Outline

Epithelial sheets (rosettes)

Cranial neural crest cell invasion

Acid-mediated invasion hypothesis (cancer)

ROSETTES

Egg-cylinder stage mouse embryo: AVE

cells migrate in a typical way to correctly

orientate A-P axis

Shankar Srinivas Aaron Smith

Bradley Joyce David Kay

Abigail Moore Ruth Baker

Tristan Rodriguez

Jacintha Sugnaseelan

Georgios Trichas

Natalia White

Vivienne Wilkins

• The AVE then induces anterior pattern in the underlying epiblast by restricting expression of posterior markers to the opposite side of the epiblast cup

• In Nodal and Cripto mutants the AVE forms but does not migrate � severe gastrulation defects and failure to develop further

During AVE migration, multi-cellular

rosettes form (discovered by Shankar

Srinivas). Why?

Rosettes form by cellular rearrangement:

• time-lapse (AVE cells in green, expressing Hex-GFP). Scale bar 50 microns

Vertex Model (Oster and Weliky, Honda,

Julicher, etc etc)

Junctional rearrangements

• We allow vertices closer than a certain

threshold distance to join together.

Vertex rearrangement

known as a T1 swap

Cell growth and mitosis

• Cells are assigned a certain volume and which

grows over time

• Cells can divide when they reach a certain

volume

Induce Migration

Comparison with experiment: I

• Reduction in mean polygon number in the Epi-VE (pre- and post-migration) agreeing with observations:

Comparison with experiment: II

• Similar trends observed – 6-sided cells relatively less frequent in Epi-VE, while 4-sided cells more frequent there

Abnormal AVE migration in ROSA26 mutant

• Has disrupted PCP [Planar Cell Polarity] and forms fewer rosettes

Migration without rosettes

Migration with or without rosettes

Conclusions

• Our simple model qualitatively captures a lot of features observed in AVE migration

• Rosettes not essential for successful AVE migration

• Rosettes are essential for ordered migration observed in vivo

• Trichas, Smith, White, Wilkins, Watanabe, Moore, Joyce, Sugnaseelan, Rodriguez, Kay, Baker, Maini, Srinivas, Multi-cellular rosettes in the mouse visceral endoderm facilitate the ordered migration of anterior visceral endoderm cells, PLoS Biology, 2012, 10(2), e1001256

MODELLING EPITHELIAL SHEETS

Cell-based modelling approaches

• Cell-centred

• Vertex Models (Oster, Honda, Davidson et etc)

• Potts Models (Glazier, Graner, Hogeweg)

• WHICH IS CORRECT??

• Chaste – Cancer, Heart And Soft Tissue Environment

• Modular

• Developed in a collaboration between Computer

Science, Oxford and CMB – lead developers David

Gavaghan, Joe Pitt-Francis, James Osborne and Alex

Fletcher

Movies (Mirams, Fletcher, PKM and Byrne, JTB, 2012)

• Individual cell movement at the discrete scale modelled using nonlinear force laws can be described by nonlinear diffusioncoefficients on the continuum scale.

• Therefore, we can (a) relate different discrete models of cell behaviour; (b) derive discrete, inter-cell force laws from previously posed diffusion coefficients

• P.J. Murray, C.M. Edwards, M.J. Tindall, PKM, Classifying general linear force laws in cell-based models via the continuum limit (Phys Rev E, 2012)

• Comparing vertex-based models (Fletcher, Osborne, PKM, Gavaghan, submitted)

Movies (Mirams, Fletcher, PKM and Byrne, JTB, 2012)

NEURAL CREST CELL MIGRATION

Migration of Cranial Neural Crest Cells

• Paul Kulesa, Rebecca McLennan, Katherine

Prather, Jason Morrison

[Stowers Institute, Kansas]

• Louise Dyson, Ruth Baker

• NC cell population is crucial for proper development of the heart and peripheral nervous systems

• Is the cellular origin of the highly aggressive cancers, melanoma and neuroblastoma

Model and manipulation

• Can a chemoattractant (VEGF) produced by the overying ectoderm be sufficient for robust invasion?

Cell invasion with one cell type with chemoattractant modelled by a

reaction-diffusion equation

Cell invasion with “leaders” and “followers”

Model Prediction and Validation

• A single chemotactic gradient with a single cell type is not a feasible mechanism. There must be at least 2 cell types – one chemotactic, one not chemotactic.

• By FACS (flow cytometry analysis) and by LCM (laser capture microdissection) show significant differences in expression of 19 out of 84 genes.

• Leading NC cells have upregulated expression of cell guidance and navigation genes (cell guidance factor recptors [EphA4]; integrins (Itgb5); MMPs [MMP2]; cadherins [Cdh7]). Trailing cells have upregulatedexpression of cadherins distinct from leading NC cells.

Loss of Trailing Cells

• Disrupts migration (maybe the effects of pushing are lost)

• No change – leaders forge ahead• Affects movement of leaders through a

global change in chemoattractant profile

Cells move more slowly due to a less directed VEGF gradient

Transplant Trailing Cells to the Front

2D is NOT 3D !!

Lead Cells Transplanted Proximally

• They maintain their ordering and migrate• May overtake the host cells as, after all,

they are the leaders

Model Prediction – nothing happens

C = cell density, N = cell number, d= fixed jump size

Variable jump size (normally distributed)

Exclusion principle

References

• McLennan, Dyson, Prather, Morrison, Baker, Maini, Kulesa, Multiscale mechanisms of cell migration during development: theory and experiment, Development, 139, 2935-2944 (2012)

• L. Dyson, P.K. Maini, R.E. Baker, Macroscopic limits of individual-based models for motile cell populations with volume exclusion, Phys. Rev. E. 86, 031903 (5 pages) (2012)

ACID-MEDIATED INVASION HYPOTHESIS

Acid-Mediated Invasion Hypothesis

• A bi-product of the glycolytic pathway is lactic acid – this lowers the extracellular pH so that it favours tumour cell proliferation AND it is toxic to normal cells.

• R.A. Gatenby and E.T. Gawslinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56, 5745-5753 (1996)

Gatenby-Gawlinski Model

Travelling waves of invasion

Experimental results(Gatenby et al, Cancer Research, 66, 5216-5223, 2006)

Therapeutics

• Add bicarbonate to neutralise the acid

Metastatic Lesions (Robey et al, Cancer Res, 69(6), 2260-2268)

• In the asymptotic limit (small parameter is tumour cell diffusion divided by lactic acid diffusion) this is Fisher’s eqn.

• Using asymptotics (some subtilies arise) --determine parameter space in which gaps occur

• McGillen, Martin, Robey, Gaffney, PKM, Riken Proceedings (Nishiura 60th

birthday) in press • McGillen, Martin, Gaffney, PKM (J. Math. Biol. to appear)

FROM MOUSE TO HUMAN

Equivalent dose less effective in humans (Du Bois height-weight

formula)

Results

• Previously unseen potentially dangerous elevation in blood pHe resulting from bicarbonate therapy in mice – confirmed by our in vivo expts

• Limited efficacy of bicarbonate in humans

• Martin, Robey, Gaffney, Gillies, Gatenby, PKM, Predicting the safety and efficacy of buffer therapy to raise tumour pHe: An intergrative modelling study, Brit. J. Cancer, 106,1280-1287 (2012)

Buffer therapy most effective

(a) in elderly patients with renal impairments

(b) in combination with proton production inhibitors (such as DCA), renal glomular filtration rate inhibitors (eg non-steroidal anti-inflammatory drugs and angiotensin-converting enzyme inhibitors) or

(c) with an alternative buffer reagent possessing an optimal pK of 7.1 – 7.2 (pK = -log(dissociation constant) – does not elevate (to dangerous levels) blood pH in the same way that bicarbonate can!!

Conclusions

(a) Used a vertex-based model for AVE migration on the mouse ectoderm to investigate the role of rosettes

(b) Used a hybrid model to study neuralcrest cell invasion

(c) Used a PDE model to study tumour cellinvasion

(d) Showed how many of these approaches lead to novel nonlinear diffusion type systems

AcknowledgementsShankar Srinivas Katherine Prather Ruth Baker Jessica McGillenBradley Joyce Jason Morrison David Kay Eamonn GaffneyAbigail Moore Ian Robey Aaron Smith Louise Dyson Tristan Rodriguez Bob Gatenby Joe Pitt-Francis Helen ByrneJacintha Sugnaseelan Bob Gillies James Osborne Philip MurrayGeorgios Trichas Katherine Prather David Gavaghan Marcus TindalNatalia White Paul Kulesa Gary Mirams Carina EdwardsVivienne Wilkins Rebecca McLennan Alex Fletcher Natasha Martin

• FundingBBRSC, EPSRC (IB, 2020 Science, DTCs), NIH-NCI